0
Research Papers

Chirality Induced by Structural Transformation in a Tensegrity: Theory and Experiment

[+] Author and Article Information
Li-Yuan Zhang

School of Mechanical Engineering,
University of Science and Technology Beijing,
Beijing 100083, China;
Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China

Zi-Long Zhao

Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China

Qing-Dong Zhang

School of Mechanical Engineering,
University of Science and Technology Beijing, Beijing 100083, China

Xi-Qiao Feng

AML & CNMM,
Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China
e-mail: fengxq@tsinghua.edu.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received December 4, 2015; final manuscript received December 25, 2015; published online January 18, 2016. Editor: Yonggang Huang.

J. Appl. Mech 83(4), 041003 (Jan 18, 2016) (7 pages) Paper No: JAM-15-1654; doi: 10.1115/1.4032375 History: Received December 04, 2015; Revised December 25, 2015

Chiral structures have many technologically significant applications in engineering. In this paper, we investigate, both theoretically and experimentally, the structural transformation from a symmetric X-shaped tensegrity to a chiral structure under uniaxial tension. When the applied tensile force exceeds a critical value, the initially achiral structure would exhibit snap-through buckling. At the critical state, the in-plane deformation mode of the tensegrity switches into an off-plane one. The critical condition of the structural transformation is provided in terms of structural parameters. An experiment was performed to validate the theoretical model. This work may not only deepen our understanding of the stability of tensegrities but also help design chiral structures for engineering applications.

FIGURES IN THIS ARTICLE
<>
Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.

References

Morris, R. E. , and Bu, X. H. , 2010, “ Induction of Chiral Porous Solids Containing Only Achiral Building Blocks,” Nat. Chem., 2(5), pp. 353–361. [CrossRef] [PubMed]
Forterre, Y. , and Dumais, J. , 2011, “ Generating Helices in Nature,” Science, 333(6050), pp. 1715–1716. [CrossRef] [PubMed]
Zhao, Z. L. , Zhao, H. P. , Wang, J. S. , Zhang, Z. , and Feng, X. Q. , 2014, “ Mechanical Properties of Carbon Nanotube Ropes With Hierarchical Helical Structures,” J. Mech. Phys. Solids, 71, pp. 64–83. [CrossRef]
Schulgasser, K. , and Witztum, A. , 2004, “ The Hierarchy of Chirality,” J. Theor. Biol., 230(2), pp. 281–288. [CrossRef] [PubMed]
Liu, Z. F. , Gandhi, C. S. , and Rees, D. C. , 2009, “ Structure of a Tetrameric MscL in an Expanded Intermediate State,” Nature, 461(7260), pp. 120–124. [CrossRef] [PubMed]
Severcan, I. , Geary, C. , Chworos, A. , Voss, N. , Jacovetty, E. , and Jaeger, L. , 2010, “ A Polyhedron Made of tRNAs,” Nat. Chem., 2(9), pp. 772–779. [CrossRef] [PubMed]
Zhao, Z. L. , Huang, W. X. , Li, B. W. , Chen, K. X. , Chen, K. F. , Zhao, H. P. , and Feng, X. Q. , 2015, “ Synergistic Effects of Chiral Morphology and Reconfiguration in Cattail Leaves,” J. Bionic Eng., 12(4), pp. 634–642. [CrossRef]
Zhao, Z. L. , Zhao, H. P. , Li, B. W. , Nie, B. D. , Feng, X. Q. , and Gao, H. J. , 2015, “ Biomechanical Tactics of Chiral Growth in Emergent Aquatic Macrophytes,” Sci. Rep., 5, p. 12610. [CrossRef] [PubMed]
Schulgasser, K. , and Witztum, A. , 2004, “ Spiralling Upward,” J. Theor. Biol., 230(2), pp. 275–280. [CrossRef] [PubMed]
Wu, J. , Hwang, K. C. , Huang, Y. , and Song, J. , 2008, “ A Finite-Deformation Shell Theory for Carbon Nanotubes Based on the Interatomic Potential—Part I: Basic Theory,” ASME J. Appl. Mech., 75(6), p. 061006. [CrossRef]
Calladine, C. R. , Luisi, B. F. , and Pratap, J. V. , 2013, “ A ‘Mechanistic’ Explanation of the Multiple Helical Forms Adopted by Bacterial Flagellar Filaments,” J. Mol. Biol., 425(5), pp. 914–928. [CrossRef] [PubMed]
Wang, J. S. , Wang, G. , Feng, X. Q. , Kitamura, T. , Kang, Y. L. , Yu, S. W. , and Qin, Q. H. , 2013, “ Hierarchical Chirality Transfer in the Growth of Towel Gourd Tendrils,” Sci. Rep., 3, p. 3102. [PubMed]
Zhao, Z. L. , Zhao, H. P. , Chang, Z. , and Feng, X. Q. , 2014, “ Analysis of Bending and Buckling of Pre-Twisted Beams: A Bioinspired Study,” Acta Mech. Sin., 30(4), pp. 507–515. [CrossRef]
Tee, Y. H. , Shemesh, T. , Thiagarajan, V. , Hariadi, R. F. , Anderson, K. L. , Page, C. , Volkmann, N. , Hanein, D. , Sivaramakrishnan, S. , Kozlov, M. M. , and Bershadsky, A. D. , 2015, “ Cellular Chirality Arising From the Self-Organization of the Actin Cytoskeleton,” Nat. Cell Biol., 17(4), pp. 445–457. [CrossRef] [PubMed]
Ye, H. M. , Wang, J. S. , Tang, S. , Xu, J. , Feng, X. Q. , Guo, B. H. , Xie, X. M. , Zhou, J. J. , Li, L. , Wu, Q. , and Chen, G. Q. , 2010, “ Surface Stress Effects on the Bending Direction and Twisting Chirality of Lamellar Crystals of Chiral Polymer,” Macromolecules, 43(13), pp. 5762–5770. [CrossRef]
Guest, S. D. , Kebadze, E. , and Pellegrino, S. , 2011, “ A Zero-Stiffness Elastic Shell Structure,” J. Mech. Mater. Struct., 6(1–4), pp. 203–212. [CrossRef]
Ji, X. Y. , Zhao, M. Q. , Wei, F. , and Feng, X. Q. , 2012, “ Spontaneous Formation of Double Helical Structure Due to Interfacial Adhesion,” Appl. Phys. Lett., 100(26), p. 263104. [CrossRef]
Gruziel, M. , Dzwolak, W. , and Szymczak, P. , 2013, “ Chirality Inversions in Self-Assembly of Fibrillar Superstructures: A Computational Study,” Soft Matter, 9(33), pp. 8005–8013. [CrossRef]
Wang, J. W. , Cao, Y. P. , and Feng, X. Q. , 2014, “ Archimedean Spiral Wrinkles on a Film-Substrate System Induced by Torsion,” Appl. Phys. Lett., 104(3), p. 031910. [CrossRef]
Cahill, K. , 2005, “ Helices in Biomolecules,” Phys. Rev. E, 72(6), p. 062901. [CrossRef]
Snir, Y. , and Kamien, R. D. , 2005, “ Entropically Driven Helix Formation,” Science, 307(5712), p. 1067. [CrossRef] [PubMed]
Wang, J. S. , Feng, X. Q. , Wang, G. F. , and Yu, S. W. , 2008, “ Twisting of Nanowires Induced by Anisotropic Surface Stresses,” Appl. Phys. Lett., 92(19), p. 191901. [CrossRef]
Wang, J. S. , Ye, H. M. , Qin, Q. H. , Xu, J. , and Feng, X. Q. , 2012, “ Anisotropic Surface Effects on the Formation of Chiral Morphologies of Nanomaterials,” Proc. R. Soc. A, 468(2139), pp. 609–633. [CrossRef]
Armon, S. , Efrati, E. , Kupferman, R. , and Sharon, E. , 2011, “ Geometry and Mechanics in the Opening of Chiral Seed Pods,” Science, 333(6050), pp. 1726–1730. [CrossRef] [PubMed]
Li, B. , Cao, Y. P. , Feng, X. Q. , and Gao, H. J. , 2012, “ Mechanics of Morphological Instabilities and Surface Wrinkling in Soft Materials: A Review,” Soft Matter, 8(21), pp. 5728–5745. [CrossRef]
Wu, Z. L. , Moshe, M. , Greener, J. , Therien-Aubin, H. , Nie, Z. H. , Sharon, E. , and Kumacheva, E. , 2013, “ Three-Dimensional Shape Transformations of Hydrogel Sheets Induced by Small-Scale Modulation of Internal Stresses,” Nat. Commun., 4, p. 1586. [CrossRef] [PubMed]
Guo, Q. , Mehta, A. K. , Grover, M. A. , Chen, W. , Lynn, D. G. , and Chen, Z. , 2014, “ Shape Selection and Multi-Stability in Helical Ribbons,” Appl. Phys. Lett., 104(21), p. 211901. [CrossRef]
Chen, Z. , 2015, “ Shape Transition and Multi-Stability of Helical Ribbons: A Finite Element Method Study,” Arch. Appl. Mech., 85(3), pp. 331–338. [CrossRef]
Kardas, D. , Nackenhorst, U. , and Balzani, D. , 2013, “ Computational Model for the Cell-Mechanical Response of the Osteocyte Cytoskeleton Based on Self-Stabilizing Tensegrity Structures,” Biomech. Model. Mechanobiol., 12(1), pp. 167–183. [CrossRef] [PubMed]
Ingber, D. E. , Wang, N. , and Stamenović, D. , 2014, “ Tensegrity, Cellular Biophysics, and the Mechanics of Living Systems,” Rep. Prog. Phys., 77(4), p. 046603. [CrossRef] [PubMed]
Motro, R. , 2003, Tensegrity: Structural Systems for the Future, Butterworth-Heinemann, London, UK.
Skelton, R. E. , and de Oliveira, M. C. , 2009, Tensegrity Systems, Springer, Dordrecht, The Netherlands.
Zhang, L. Y. , Li, Y. , Cao, Y. P. , Feng, X. Q. , and Gao, H. J. , 2012, “ Self-Equilibrium and Super-Stability of Truncated Regular Polyhedral Tensegrity Structures: A Unified Analytical Solution,” Proc. R. Soc. A, 468(2147), pp. 3323–3347. [CrossRef]
Zhang, J. Y. , and Ohsaki, M. , 2012, “ Self-Equilibrium and Stability of Regular Truncated Tetrahedral Tensegrity Structures,” J. Mech. Phys. Solids, 60(10), pp. 1757–1770. [CrossRef]
Zhang, L. Y. , Li, Y. , Cao, Y. P. , Feng, X. Q. , and Gao, H. J. , 2013, “ A Numerical Method for Simulating Nonlinear Mechanical Responses of Tensegrity Structures Under Large Deformations,” ASME J. Appl. Mech., 80(6), p. 061018. [CrossRef]
Fraternali, F. , Carpentieri, G. , and Amendola, A. , 2015, “ On the Mechanical Modeling of the Extreme Softening/Stiffening Response of Axially Loaded Tensegrity Prisms,” J. Mech. Phys. Solids, 74, pp. 136–157. [CrossRef]
Zhang, L. , Lu, M. K. , Zhang, H. W. , and Yan, B. , 2015, “ Geometrically Nonlinear Elasto-Plastic Analysis of Clustered Tensegrity Based on the Co-Rotational Approach,” Int. J. Mech. Sci., 93, pp. 154–165. [CrossRef]
Yuan, X. F. , Peng, Z. L. , Dong, S. L. , and Zhao, B. J. , 2008, “ A New Tensegrity Module—‘Torus',” Adv. Struct. Eng., 11(3), pp. 243–251. [CrossRef]
Korkmaz, S. , Ali, N. B. H. , and Smith, I. F. C. , 2011, “ Determining Control Strategies for Damage Tolerance of an Active Tensegrity Structure,” Eng. Struct., 33(6), pp. 1930–1939. [CrossRef]
Sunny, M. R. , Sultan, C. , and Kapania, R. K. , 2014, “ Optimal Energy Harvesting From a Membrane Attached to a Tensegrity Structure,” AIAA J., 52(2), pp. 307–319. [CrossRef]
Luo, Y. Z. , Xu, X. , Lele, T. , Kumar, S. , and Ingber, D. E. , 2008, “ A Multi-Modular Tensegrity Model of an Actin Stress Fiber,” J. Biomech., 41(11), pp. 2379–2387. [CrossRef] [PubMed]
Stamenović, D. , and Ingber, D. E. , 2009, “ Tensegrity-Guided Self Assembly: From Molecules to Living Cells,” Soft Matter, 5(6), pp. 1137–1145. [CrossRef]
Scarr, G. , 2011, “ Helical Tensegrity as a Structural Mechanism in Human Anatomy,” Int. J. Osteopath. Med., 14(1), pp. 24–32. [CrossRef]
Fraternali, F. , Senatore, L. , and Daraio, C. , 2012, “ Solitary Waves on Tensegrity Lattices,” J. Mech. Phys. Solids, 60(6), pp. 1137–1144. [CrossRef]
Zhang, L. Y. , Zhang, C. , Feng, X. Q. , and Gao, H. J. , 2016, “ Snapping Instabilities in a Prismatic Tensegrity Under Torsion,” Appl. Math. Mech. Engl. Ed., 37 (in press).
Zhang, L. Y. , and Xu, G. K. , 2015, “ Negative Stiffness Behaviors Emerging in Elastic Instabilities of Prismatic Tensegrities Under Torsional Loading,” Int. J. Mech. Sci., 103, pp. 189–198. [CrossRef]
Tibert, A. G. , and Pellegrino, S. , 2003, “ Review of Form-Finding Methods for Tensegrity Structures,” Int. J. Space Struct., 18(4), pp. 209–223. [CrossRef]
Zhang, L. Y. , Li, Y. , Cao, Y. P. , and Feng, X. Q. , 2014, “ Stiffness Matrix Based Form-Finding Method of Tensegrity Structures,” Eng. Struct., 58, pp. 36–48. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Configuration of X-shaped tensegrity in the absence of external load. The spheres, thin lines, and thick lines denote the nodes, strings, and bars, respectively.

Grahic Jump Location
Fig. 2

Two possible deformation modes of X-shaped tensegrity under uniaxial tension: (a) in-plane mode (α=0) and (b) off-plane mode (0<α<π), with α denoting the relative twist angle between the top and bottom strings

Grahic Jump Location
Fig. 3

Curves of the uniaxial force f¯, incremental stiffness Δf¯/Δλ¯, and twist angle α with respect to relative elongation λ¯ for X-shaped tensegrity with k¯s2=1, k¯b=100, L¯s2=1, and L¯b=1.5

Grahic Jump Location
Fig. 4

Curves of the critical force f¯cr and critical elongation λ¯cr with respect to the axial stiffness k¯s2 of the horizontal strings for an X-shaped tensegrity with k¯b=100, L¯s2=1, and L¯b=1.5. The inset shows the details of f¯cr – k¯s2 and λ¯cr – k¯s2 curves in a subrange.

Grahic Jump Location
Fig. 5

Curves of the critical force f¯cr and critical elongation λ¯cr with respect to the axial stiffness k¯b of the bars for an X-shaped tensegrity with k¯s2=1, L¯s2=1, and L¯b=1.5. The inset shows the details of f¯cr – k¯b and λ¯cr – k¯b curves in a subrange.

Grahic Jump Location
Fig. 6

Curves of the critical force f¯cr and critical elongation λ¯cr with respect to the natural length L¯s2 of the horizontal strings for an X-shaped tensegrity with k¯s2=1, k¯b=100, and L¯b=1.5

Grahic Jump Location
Fig. 7

Curves of the critical force f¯cr and critical elongation λ¯cr with respect to the natural length L¯b of the bars for an X-shaped tensegrity with k¯s2=1, k¯b=100, and L¯s2=1. The inset shows the details of f¯cr – L¯b and λ¯cr – L¯b curves in a subrange.

Grahic Jump Location
Fig. 8

X-shaped tensegrity sculpture: (a) planar configuration without external load and (b) chiral configuration under an external load exceeding the critical force

Grahic Jump Location
Fig. 9

Variations of the uniaxial force f¯ versus relative elongation λ¯ for the tensegrity in Fig. 8(a) with different vertical springs: (a) ks1=0.116 N/mm and Ls1=158 mm and (b) ks1=0.122 N/mm and Ls1=148 mm. The symbols are the experimental results under four loading–unloading processes, and the lines are the theoretical results calculated from Eqs. (17)(20).

Grahic Jump Location
Fig. 10

Strip structure made of five X-shaped tensegrity cells with different critical forces: (a) curves of uniaxial force f̂ and total twist angle A with respect to relative elongation λ̂ and (b) configuration evolution of the structure under uniaxial tensile force

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In